January 19, 2013

Logic Part 4: How To Make Truth Tables

Truth tables are an important tool for evaluating statements and arguments. We can create our own truth tables using following steps:

Translate statements of ordinary language.

Break all complex statements into smaller parts.

Determine how many columns are required.

Determine how many rows are required.

Determine the truth values of statement letters.

Determine the truth values of complex statements.

I will illustrate how to follow these steps by using an example. In particular, I will show how we can make a truth table of an argument to find out if the argument is logically valid.

Step 1: Translate statements of ordinary language.

We can make a truth table of one or more statements. Arguments can be presented on a truth table by presenting all the statements from the argument. We will make a truth table of an argument, but that requires that we first translate an argument of ordinary language into propositional logic.

Consider the following argument written in English:

If Mary will either go to the store or stay at home, then Mark will go to the store and buy milk. Mary will not go to the store or stay at home. Therefore, it’s not the case that Mark will go to the store and buy milk.

We can identify the premises and conclusion:

Premise 1: If Mary will either go to the store or stay at home, then Mark will go to the store.

Premise 2: Mary will not go to the store or stay at home.

Conclusion: It’s not the case that Mark will go to the store.

We can then replace each statement with a propositional variable and translate these statements into propositional logic:

(A ∨ B) → C

¬(A ∨ B)

¬C

A: Mary will go to the store.

B: Mary will stay at home.

C: Mark will go to the store.

Step 2: Break all complex statements into smaller parts.

We can break statements into smaller parts by removing logical connectives. Each logical connective must be removed one at a time until only statement letters are left. Whenever a logical connective is removed, it should be the one with the least amount of parentheses around it.

First statement

Look at the first propositional statement of the argument: (A ∨ B) → C

First remove the conditional (→) to create the following simpler statement and statement letter: A ∨ B, C

We still need to break “A ∨ B” into smaller parts. Next remove the disjunction (∨) to get the following two statement letters: A, B

Second statement

Look at the second propositional statement of the argument: ¬(A ∨ B)

First we need to remove the negation (¬) to get the following simpler statement: A ∨ B

We still need to break “A ∨ B” into smaller parts. We need to remove the disjunction to get the following statement letters: A, B

Third statement

Look at the final statement of the argument: ¬C

We need to remove the negation to get the following statement letter: C

Overview

A list of all statements including the smaller parts include the following:

(A ∨ B) → C

C

A ∨ B

A

B

¬(A ∨ B)

¬C

Step 3: Determine how many columns are required.

Columns are the vertical areas of the truth table. We need one column for each statement of the argument and the smaller parts of those statements. If you look at the list of all the statements and smaller parts, you will see that there are seven statements that the table will need.

The top row of the truth table will contain these statements and look like the following:

A

B

C

A ∨ B

(A ∨ B) → C

¬(A ∨ B)

¬C

Step 4: Determine how many rows are required.

Rows are the horizontal areas of the truth table. A number of rows is required in order to determine every possible truth value, which depends on the number of statement letters used:

Step 5: Determine the truth values of statement letters.

The last statement letter alternates from being true and false every other row starting with true (T, F, T, F…). The second to last statement letter alternates from being true and false every two rows (T, T, F, F, etc.) The third from last statement letter alternates from being true and false every four rows (T, T, T, T, F, F, F, F, etc.) The first statement of each table should be true in the first half of the rows and false on the bottom half.

We can now determine when A, B, and C are true or false:

A

B

C

A ∨ B

(A ∨ B) → C

¬(A ∨ B)

¬C

T

T

T

T

T

F

T

F

T

T

F

F

F

T

T

F

T

F

F

F

T

F

F

F

Step 6: Determine the truth values of complex statements.

There are truth tables that determine when a statement with a logical connective is true or false. We can summarize these tables as saying the following:

Conjunction (∧) – “p ∧ q” is only true when p is true and q is true.

Negation (¬) – “¬p” has the opposite truth value of “p.”

Disjunction (∨) – “p ∨ q” is only false when both p and q are false.

Conditional (→) – “p → q” is only false when p is true and q is false.

Equivalence (↔) – “p ↔ q” is only true when p and q have the same truth value.

The first statement with a connective that we need to determine the truth values for is “A ∨ B.” It will only be false when both A and B are false because it’s a disjunction:

A

B

C

A ∨ B

(A ∨ B) → C

¬(A ∨ B)

¬C

T

T

T

T

T

T

F

T

T

F

T

T

T

F

F

T

F

T

T

T

F

T

F

T

F

F

T

F

F

F

F

F

The second statement we need the truth values for is “(A ∨ B) → C.” It will only be false when “ A ∨ B” is true and “C” is false because it’s a conditional:

A

B

C

A ∨ B

(A ∨ B) → C

¬(A ∨ B)

¬C

T

T

T

T

T

T

T

F

T

T

T

F

T

T

T

T

F

F

T

F

F

T

T

T

T

F

T

F

T

F

F

F

T

F

T

F

F

F

F

T

The third statement we need to find truth values for is “¬(A ∨ B).” It will have the opposite truth values as “A ∨ B” because it’s a negation:

A

B

C

A ∨ B

(A ∨ B) → C

¬(A ∨ B)

¬C

T

T

T

T

T

F

T

T

F

T

T

F

T

F

T

T

T

F

T

F

F

T

F

F

F

T

T

T

T

F

F

T

F

T

F

F

F

F

T

F

T

T

F

F

F

F

T

T

The fourth and final statement we need to find truth values for is “¬C.” It will have the opposite truth values as “C” because it’s a negation:

A

B

C

A ∨ B

(A ∨ B) → C

¬(A ∨ B)

¬C

T

T

T

T

T

F

F

T

T

F

T

T

F

T

T

F

T

T

T

F

F

T

F

F

T

F

F

T

F

T

T

T

T

F

F

F

T

F

T

F

F

T

F

F

T

F

T

T

F

F

F

F

F

T

T

T

The reason we usually want to make a truth table of an argument is to find out if it’s logically valid. It is logically valid if it’s impossible for it to have true premises and a false conclusion at the same time. It is logically invalid if it is possible for it to have true premises and a false conclusion at the same time.

We can look at the truth table to find out if it’s valid. Look at all the rows where both premises are true and see if the conclusion is ever false on those rows. If so, it’s invalid. Otherwise it’s valid.

The premises are the following:

(A ∨ B) → C

¬(A ∨ B)

The conclusion is the following:

¬C

The premises are only true on the same row on the bottom two rows, and the conclusion is false on one of those rows. Therefore, the argument is logically invalid: